Related papers: On basic $r$-ball polyhedra
Following a review of related results in rigidity theory, we provide a construction to obtain generically universally rigid frameworks with the minimum number of edges, for any given set of n nodes in two or three dimensions. When a set of…
We study approximations of smooth convex bodies by random ball-polytopes. We examine the following probability model: let $K\subset{\bf R}^d$ be a convex body such that $K$ slides freely in a ball of radius $R>0$ and has $C^2$ smooth…
Let P be a locally finite disk pattern on the complex plane C whose combinatorics are described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function \Theta:E\to [0,\pi/2] on…
We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces…
This article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the…
A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and…
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume…
Using a quartic surface and its rational curves we can give an infinite number of integer hexahedra; these are 6 sided 3d solids, each face a trapezoid, with all sides and diagonals having intger lengths.
A bar framework determined by a finite graph $G$ and configuration $\bf p$ in $d$ space is universally rigid if it is rigid in any ${\mathbb R}^D \supset {\mathbb R}^d$. We provide a characterization of universally rigidity for any graph…
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with…
We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the…
Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating…
We give a new upper bound $K_+$ on the number of totally elastic collisions of $n$ hard spheres with equal radii and equal masses in $R^d$. Our bound satisfies $\log K_+ \leq c(d) n \log n$.
An orthant polyhedron is a polyhedron with $m$ hyperfaces, that could be realized as a section of the $m$-dimensional non-negative orthant. We classify all 2-dimensional orthant polyhedra and provide some partial results towards the…
For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…
A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in…
Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction…
We solve a very classical problem motivated by considerations in quantum gravity: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms…
We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as…
We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for an odd…